CHAPTER XI.

THE DISTANCES AND DIMENSIONS OF THE HEAVENLY BODIES.

PARALLAX.—The problem of determining the distance of a heavenly body resolves itself into a measurement of its _parallax_, that is, of the apparent change of its position brought about by a change in the situation of an observer. If one be seated in a room, about a yard from a window, a very simple experiment may be made to illustrate the meaning of this term. Closing one eye, the observer will see a vertical line, such as the partition between two panes, projected upon some particular part of an opposite building; when the other eye is used the line will apparently be displaced, and the nearer one is to the window the greater will be the displacement or parallax. As the heavenly bodies are so far away, each of our eyes sees them in the same directions. Indeed, the stars are so distant that to _all_ persons situated on our planet their apparent positions are identical. With the members of the solar system, however, the case is different; the earth has an appreciable size as seen from them, so that when viewed from different parts of the earth they will not appear in exactly the same part of the heavens.

The earth’s rotation changes the relation of an observer’s position with regard to a heavenly body in pretty much the same way as a change in his actual position on the globe. When an object in the zenith is observed, it will appear in precisely the same part of the sky as if it were seen from the centre of the earth, but as it approaches the horizon it will be displaced. Hence the term _diurnal parallax_, meaning the displacement of a heavenly body depending upon the observer’s position as affected by the earth’s rotation. Taking it in its general astronomical sense, the parallax of a heavenly body is the angle between the two lines which join it to the observer and to the centre of the earth respectively. Thus, in Fig. 38, let O be an observer, Z his zenith, and C the centre of the earth; then the parallax of a body S is the angle O S C. As the observer’s position is changed to O′ by the earth’s rotation, the parallactic angle is increased to O′ S C. If S be on the horizon, that is, when O′ C is perpendicular to O′ S, the parallax is a maximum, and is then called the horizontal parallax. The _horizontal parallax_ of a body is therefore the greatest angle subtended by the earth’s radius as seen from the body. We have seen, however, that the earth’s radius is not of the same length in all parts, and it is therefore necessary to specify more particularly which radius is in question. The standard adopted is the equatorial radius, and, when this is employed, our greatest parallactic angle is called the _equatorial horizontal parallax_.

[Illustration:

FIG. 38.—_Parallax of a Heavenly Body._ ]

In the case of all the heavenly bodies the parallaxes are very small; that of the moon averages about 57′, while that of the nearest planet does not exceed 40″. The parallax of a body evidently diminishes as the distance increases.

DISTANCE DEDUCED FROM PARALLAX.—When the parallax of a heavenly body has been determined, it becomes a simple matter to calculate the corresponding distance; thus, in Fig. 38, the distance C O′ represents the earth’s equatorial radius, O′ S C is the equatorial horizontal parallax, C O′ S is a right angle, and the required distance is C S. By a simple trigonometrical rule this distance is the earth’s radius divided by the sine of the parallax. In the case of a small angle, the sine is very nearly equal to the angle itself divided by the angle corresponding to an arc of a circle equal in length to the radius. As there are 206,265 seconds in an arc equal to the radius, the sine of a small angle may be taken as the angle itself, expressed in seconds, divided by this number. Thus, if _p_ be the equatorial horizontal parallax of an object reckoned in seconds of arc,

Distance = (earth’s equatorial radius)/(sine _p_) = (206,265 × earth’s equatorial radius)/(_p_)

We shall see presently that the average parallax of the sun is 8″·80, and its average distance, as derived from the application of this formula, is accordingly about 92,790,000 miles.

DIAMETERS.—It is a familiar fact that the further an object is removed from us the smaller it appears. The ascent of a balloon at once suggests itself as an excellent example. It is necessary, therefore, to distinguish very carefully between the apparent and the true size of an object. A halfpenny at a distance of nine feet from the eye will just cover the moon if the line of sight be directed towards that body, but we should not say the moon is the size of a halfpenny, because we know perfectly well that a disc twice the size would produce just the same appearance if removed to double the distance. Apparent size must, accordingly, be reckoned in angular measure, and we say, for example, that the moon has an apparent diameter of a little more than half a degree.

When the angular diameter and distance have both been measured, the real diameter, in miles, can at once be deduced by a simple inversion of the process of determining the distance of an object from its known parallax. Thus, in Fig. 39 let A B represent the moon or other heavenly body, and E the centre of the earth. The angle M E A is the angular semi-diameter, and E M the required distance; then, since the angle E A M is a right angle,

A M = M E × sine M E A

That is,

Semi-diameter in miles = distance in miles × sine of angular semi-diameter.

Or,

Diameter = twice the distance × sine of angular semi-diameter.

[Illustration:

FIG 39.—_Diameter of a Heavenly Body._ ]

Since the apparent diameters are always small, the sine may be taken as equal to the circular measure; that is, the number of seconds which the angle contains divided by 206,265.

DISTANCE AND SIZE OF THE MOON.—If the moon were a fixed body outside the earth, its parallax could be easily determined by a single observer, who, in that case, would note the apparent displacement produced by his rotation. It has, however, a very complex movement, and it is therefore difficult to separate the real change of position from the parallactic change. The best method is one in which two observers, far removed from each other, can observe the moon’s position at nearly the same instant, so that the effect of its movement is very small and can be sufficiently allowed for. A necessary consequence of this condition is that the two observers should be placed as nearly as possible on the same meridian. Observations with the object of determining the lunar parallax have accordingly been made at Greenwich and the Cape of Good Hope. From the known positions of these places and the size of the earth, the distance between them is very accurately known, and this serves as a base line in a triangulation of the moon.

[Illustration:

FIG. 40.—_Measurement of the Moon’s Distance._ ]

If G and C, in Fig. 40, represent Greenwich and the Cape respectively, the celestial equators at the two places will be in the directions G E and C E. M being the moon, its declination, as measured at G, will be the angle M G E, and as measured at C it will be the angle M C E′. Since G E is parallel to C E′, the difference of these declinations (when both are north declinations, as in the diagram) will be the value of the parallactic angle G M C, which is about 1½°. From these data it is easy to calculate the distance of the moon either from Greenwich, the Cape, or the earth’s centre. In this way the distance of the moon is found at some particular moment, and the additional knowledge of the shape of its orbit enables us to determine the semi-major axis of the orbit, which is nothing more than the average or mean distance of the moon. The mean equatorial horizontal parallax of the moon is 3,422″·5, and the corresponding mean distance from the earth is 238,855 miles.

The average apparent diameter of the moon, as it would appear from the centre of the earth, is 31′ 7″, from which it results by the method already stated that the true diameter is 2,162 miles.

The apparent diameter of the moon is affected by the observer’s position upon the earth, as well as by the situation of the moon in its orbit. An observer to whom the moon is directly overhead is nearly 4,000 miles nearer to it than another observer who has it on his horizon. Tables have accordingly been drawn up to indicate the _augmentation_ of the moon’s apparent diameter as it rises above the horizon. The greatest possible apparent diameter is about 36″.

Everyone must have noticed that when the moon is rising or setting, it looks much larger than when it is high up in the sky, an appearance which does not seem to accord with the fact that its measured angular diameter is least when on the horizon. It is evident, however, that the seeming increase of size is a subjective phenomenon, due to our incapacity to correctly judge distances.

RELATIVE DISTANCES OF PLANETS.—The relative distances of the planets from the sun were found long before any of the actual distances were known with any reasonable degree of accuracy. Kepler discovered the relation which exists between these distances, and expressed it in his third or harmonic law, which states that “the squares of the periodic times of the planets are proportional to the cubes of their mean distances from the sun.”

In the case of the interior planets, the angles of greatest elongation furnish the means of finding their distances from the sun as compared with that of the earth. Thus, if V in Fig. 41 represents Venus, E the earth, and S the sun, the angle E V S is a right angle when Venus is at greatest elongation. The observed value of the angle S E V is 46°, and this definitely determines the shape, though not the size, of the triangle S E V. The distance of Venus from the sun, S V, is thus found to be 0·72 times the distance of the earth from the sun, S E. If Venus be at inferior conjunction, that is, at V′, its distance from the sun will be represented by 72, if the earth’s distance from the sun be denoted by 100.

This method can also be applied in the case of Mercury, but as the orbit is so eccentric, it is necessary to take the average of a large number of greatest elongation angles.

The process of determining the relative distance of an exterior planet, such as Jupiter, is a little more complex, but involves no considerable difficulties.

[Illustration:

FIG. 41.—_Relative Distance of Venus._ ]

There is a curious relationship between the relative distances of the planets, which is commonly known as _Bode’s law_. A series of figures, 0, 3, 6, 12, 24, 48, 96, 192, 384, each, with the exception of the second, being double the preceding one, is written down, and the number 4 added to each. Then the resulting numbers approximately represent the relative distances of the planets from the sun. Thus:—

4 7 10 16 28 52 100 196 388 Mercury Venus Earth Mars Asteroids Jupiter Saturn Uranus Neptune

It is interesting to note that this law was announced in 1772, when the asteroids and the planets Uranus and Neptune were still unknown, so that there was a break in the series corresponding to the number 28. The discovery of Uranus in 1781, and the fact that its distance agreed roughly with Bode’s law, strengthened the conviction that an unknown planet revolved round the sun in an orbit between those of Mars and Jupiter. An association of astronomers was then formed to search systematically for the missing planet; but the actual discovery was made in 1801 by Piazzi, the Sicilian astronomer, who had not joined the association. The new planet was a very small one, and its discovery was rapidly followed by the detection of several others. At the present time, more than 400 of these asteroids, or minor planets, are known, and their average distance fits in very well with Bode’s law.

THE SUN’S DISTANCE.—One of the grandest problems which astronomical science requires us to solve is the determination of the sun’s distance. Starting with a knowledge of the earth’s dimensions, the subsequent measurement of the sun’s distance enables us to get a clear idea of the scale, not only of the solar family to which we ourselves belong, but of the whole sidereal universe. No wonder then that a vast amount of astronomical energy has been expended on this investigation.

The problem, however, is beset with many practical difficulties, and the greatest possible skill is required to cope with it. In the first place, the parallax of the sun is so small that the method employed for the moon fails, and it can only be determined by indirect means.

We have already seen that the constant of aberration gives us a means of determining the size of the earth’s orbit, and consequently the distance of the sun. When proper allowance is made for the eccentricity of the orbit, this method is a very valuable one.

Other methods which have been employed depend upon the measurement of the parallax of one of the nearer planets, from which the distances of all the planets, including the earth, from the sun, can be found from our previous knowledge of the relative distances. Mars and some of the asteroids have been thus utilised at their oppositions, and Venus when at inferior conjunction.

[Illustration:

FIG. 42.—_The Parallax of Mars._ ]

The parallax of Mars can be determined in the same way as that of the moon, either by concerted observations at two distant places, or by a single observer who utilises the earth’s rotation to provide him with a base line. The actual measurements do not consist of direct estimations of the right ascension and declination of the planet, but of its angular distances from stars among which it appears, the measurements being made with micrometers or heliometers. In this way certain errors due to refraction, etc., are minimised. To take an extreme case, let the planet M (Fig. 42) be rising to an observer at O; it will then be seen in the direction O M, while a neighbouring star will be seen along the line O S. After twelve hours the rotation of the earth will have carried our observer to O′, and he will now see the planet in the direction O′ M, while the star will remain in the same direction, O′ S′. In each case he would measure the angle separating the planet from the star, and would thus obtain the values of the angles S O M and S′ O′ M, which, in the case shown in the diagram, would be together equal to the angle O M O′. When corrected for the observer’s latitude, and for the planet’s change of place in the interval, the equatorial horizontal parallax of Mars would be determined. Then the distance of Mars from the earth would be known, and at opposition this is the difference between the distances of the earth and of Mars from the sun; the ratio between the latter is already known, and their actual distances at once follow.

[Illustration:

FIG. 43.—_The Transit of Venus._ ]

TRANSIT OF VENUS.—The planet Venus at inferior conjunction is near enough to the earth to have a considerable parallax, but the method employed in the case of Mars cannot be used, as the planet is not visible when between us and the sun, except on the very rare occasions when it transits across the sun’s disc. When a transit occurs, the distance of the planet from the earth can be measured in essentially the same way as that of Mars at opposition, when two observers work together. The difference is that the apparent place of the planet is referred to the sun’s disc instead of to neighbouring stars. Suppose the conditions to be as represented in Fig. 43, E being the earth, V the planet, and S the sun. Two observers on the earth, at _a_ and _b_, will see the planet projected on different parts of the sun’s disc. If we at first regard them as being at rest, the observer at _b_ would see the planet cross the sun along the line C D, while to the one at _a_ it would appear to cross the line F G. The times of crossing would, under the assumed conditions, depend upon the orbital velocity of Venus, and a measure of these times at the two stations would determine the relative lengths of the chords C D and F G. We already know that the distance of Venus from the sun is to its distance from the earth at inferior conjunction in the proportion 72 to 28. (See p. 145.) The rectilinear distance between the two places is also known, and the distance _x y_ between the chords is ⁷²⁄₂₈ of that from _a_ to _b_, whatever the actual distance of the sun may be. We thus know the ratio of the lengths of two parallel chords, and the distance between them in miles, from which it is a simple matter to find the diameter of the sun’s disc in miles. The angular diameter of the sun is measured with a transit instrument, and to find the sun’s distance we have simply to calculate the distance at which a body of known size subtends a known angle.

We have supposed the observers at rest, but they are in reality carried forward by the earth’s orbital motion, and are turned about the earth’s axis. The first of these movements will affect both observers in the same degree, and will simply lengthen the duration of the transit. The effect of rotation, however, depends upon the position of the sun and planet, with regard to the observer’s meridian. At sunrise, an observer is carried by the rotation of the earth almost directly towards the sun, while at sunset he is carried away from it. The rate at which the planet traverses the sun’s disc would, therefore, be little affected by the earth’s rotation at sunrise or sunset. About mid-day, however, the effect of the earth’s rotation is to accelerate the apparent motion of the planet, and to shorten the time of transit. If the beginning of the transit be observed at sunset, and the end soon after sunrise, as it may well be in high latitudes, the duration of the transit is retarded by the earth’s rotation. Corrections for rotation, however, are not difficult to apply.

In this method of observing a transit of Venus, which was suggested by Halley, when it was impossible that he would live to see it carried out, the places of observation must be widely separated in latitude, and the beginning and end of the transit must both be observed.

Another method of utilising a transit of Venus is known as Delisle’s method. In this case the two stations are near the Equator, and each observer notes the Greenwich time of internal contact, when the planet fully enters upon the sun’s disc.

Owing to various causes, chief among which is the so-called “black drop,” the time of ingress and egress cannot be actually recorded with the desired degree of accuracy, and the transit Venus is no longer looked upon as the best method of determining the distance which separates us from the sun.

Some of the results which have been obtained for the solar parallax are as follows:—

Transit of Venus, 1874, contact observations, 8″·859 „ „ „ photographs, 8·859 „ „ 1882, contact observations, 8·824 „ „ „ photographs, 8·842 Gill’s observation of Mars, 1877, 8·780 Galle’s „ Flora, 1873, 8·873 Gill’s „ Juno, 1874, 8·765 „ „ minor planets, 1896, 8·80

From a discussion of all the available data, Professor Harkness considers the most probable value of the solar parallax to be 8″·80905, with a probable error of 0·00567″. Turning this into miles, we find the distance of the sun to be 92,796,950 miles, and this is in all probability not more than 60,000 miles in error. This agrees very closely with Dr. Gill’s latest value, which has been accepted by the superintendents of the British and American nautical almanacs.

THE SUN’S DIAMETER.—The real diameter of the sun is found from the parallax, and its mean angular diameter in the manner already explained (p. 142). Taking the distance as 92,780,000 miles, and the mean apparent semi-diameter as 962″, we have

Sun’s diameter = (2 × 92,780,000 × 962)/(206,265) = 865,400 miles.

The sun’s diameter is the same in all directions, so far as our measurements give any information on the point, so that there is no appreciable polar flattening corresponding to that of the earth and some of the other planets. This result is what we should expect from the relatively slow rate at which the sun turns upon its axis.

DISTANCES AND DIAMETERS OF PLANETS.—It has already been pointed out that our knowledge of the relative distances of the planets from the sun enables us to determine their absolute distances when the distance of one of them has been ascertained. In this way the determination of the earth’s distance leads us to those of the other planets.

Our additional knowledge of the planetary orbits further permits the calculation of the distance of any planet from the earth at a stated time. If, then, the angular diameter of a planet be measured with a micrometer attached to a telescope, the absolute diameter in miles can be determined in the same way as that of the sun or moon.

To take an actual example, the equatorial angular diameter of the globe of Saturn, as measured by Prof. Barnard with the great telescope of the Lick Observatory on April 14, 1895, was 19″·4. It was then computed that if the observation had been made from the sun this would have been reduced to 17″·9. The distance of Saturn from the sun being 9·5388 times the earth’s distance, it results from this measurement that the true equatorial diameter of the ball of Saturn is 76,500 miles. A number of independent measures made at intervals from March to July gave an average value of 76,470 miles for the diameter.